Question

Check that the functions are inverses. $$ f(x)=\frac{x}{4}-\frac{3}{2} \text { and } g(t)=4\left(t+\frac{3}{2}\right) $$

Short answer

Answer: Yes, the functions f(x) and g(t) are inverses of each other since composing the functions in both orders results in the identity functions (f(g(t)) = t and g(f(x)) = x).

Step-by-step solution

Compose f of g in variable t

To compute f(g(t)), we substitute g(t) into the f function where there is x. $$ f(g(t)) = f\left(4\left(t+\frac{3}{2}\right)\right) $$

Apply f to the expression inside the parentheses

Now, we must replace the x with the expression from g(t) inside the f function to obtain f(g(t)). $$ f(g(t)) = \frac{4\left(t+\frac{3}{2}\right)}{4}-\frac{3}{2} $$

Simplify f(g(t))

We have to simplify the expression to check if we get the identity function in terms of t. $$ f(g(t)) = t+\frac{3}{2} -\frac{3}{2}=t $$

Compose g of f in variable x

Similarly, to compute g(f(x)), substitute f(x) into the g function where there is t. $$ g(f(x)) = g\left(\frac{x}{4}-\frac{3}{2}\right) $$

Apply g to the expression inside the parentheses

Now, replace t with the expression from f(x) inside the g function to obtain g(f(x)). $$ g(f(x)) = 4\left(\frac{x}{4}-\frac{3}{2}+\frac{3}{2}\right) $$

Simplify g(f(x))

Lastly, simplify the expression to check if we get the identity function in terms of x. $$ g(f(x)) = x $$ The compositions of the functions resulted in the identity functions for both cases (f(g(t)) = t and g(f(x)) = x). Therefore, the functions f(x) and g(t) are inverses of each other.

Key concepts

Function Composition

Function composition is a way of combining two functions to form a single function. When we talk about composing two functions, say \( f \) and \( g \), we are essentially plugging one function into another. This means that for two functions \( f(x) \) and \( g(t) \), the composition of \( f \) and \( g \) is represented as \( f(g(t)) \). Here, you replace every occurrence of \( x \) in \( f(x) \) with \( g(t) \).

For example, if in one function \( g(t) = 4(t + \frac{3}{2}) \) and the other function \( f(x) = \frac{x}{4} - \frac{3}{2} \), we substitute \( g(t) \) into \( f(x) \), resulting in an expression we need to verify as the identity function.

• Introduce top-down substitution where the output of the inner function becomes the input of the outer function.
• Check whether composing gives back the original input value.
• Using function composition can help verify if two functions are inverses.

Identity Function

An identity function is a special type of function that returns its input unchanged. It's expressed as \( I(x) = x \) or \( I(t) = t \) depending on the variable used. The goal of checking inverse functions is essentially to get from one function back to just the variable, confirming they are inverses.

In our exercise, when composing \( f(g(t)) \), if the result is \( t \), and if \( g(f(x)) = x \), then \( f \) and \( g \) are inverses. This is because, through composition, they bring us back to the original input.

• The composition of two inverse functions results in the identity function (\( I(x) = x \) or \( I(t) = t \)).
• It's a proof that generally involves simplification where nothing but the original variable remains.
• An identity function is central in understanding inverse functions.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying an expression to achieve a desired form or outcome. In verifying inverse functions, it’s crucial to algebraically simplify each composition, \( f(g(t)) \) and \( g(f(x)) \).

For example, starting with \( f(g(t)) = \frac{4(t+\frac{3}{2})}{4} - \frac{3}{2} \), we simplify step-by-step:

• First, simplify the fraction by cancelling out terms: \( \frac{4 \cdot ...}{4} \).
• Next, simplify \( t+\frac{3}{2} - \frac{3}{2} \) to just \( t \).

These manipulations help show how each step confirms the identity function.

Similarly, for \( g(f(x)) \), when starting with \( 4\left(\frac{x}{4}-\frac{3}{2}+\frac{3}{2}\right) \), we simplify:

• Add and subtract fractions to eliminate terms.
• It's vital to maintain accuracy in each step to ensure the functions are truly inverses.

Algebraic manipulation is vital in concluding whether the composition results in something as simple as the input variable, confirming the identity relationship.